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<li><a class="reference internal" href="#">Robust covariance estimation and Mahalanobis distances relevance</a></li>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-covariance-plot-mahalanobis-distances-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="robust-covariance-estimation-and-mahalanobis-distances-relevance">
<span id="sphx-glr-auto-examples-covariance-plot-mahalanobis-distances-py"></span><h1>Robust covariance estimation and Mahalanobis distances relevance<a class="headerlink" href="#robust-covariance-estimation-and-mahalanobis-distances-relevance" title="Permalink to this headline">¶</a></h1>
<p>An example to show covariance estimation with the Mahalanobis
distances on Gaussian distributed data.</p>
<p>For Gaussian distributed data, the distance of an observation
<span class="math notranslate nohighlight">\(x_i\)</span> to the mode of the distribution can be computed using its
Mahalanobis distance: <span class="math notranslate nohighlight">\(d_{(\mu,\Sigma)}(x_i)^2 = (x_i -
\mu)'\Sigma^{-1}(x_i - \mu)\)</span> where <span class="math notranslate nohighlight">\(\mu\)</span> and <span class="math notranslate nohighlight">\(\Sigma\)</span> are
the location and the covariance of the underlying Gaussian
distribution.</p>
<p>In practice, <span class="math notranslate nohighlight">\(\mu\)</span> and <span class="math notranslate nohighlight">\(\Sigma\)</span> are replaced by some
estimates.  The usual covariance maximum likelihood estimate is very
sensitive to the presence of outliers in the data set and therefor,
the corresponding Mahalanobis distances are. One would better have to
use a robust estimator of covariance to guarantee that the estimation is
resistant to “erroneous” observations in the data set and that the
associated Mahalanobis distances accurately reflect the true
organisation of the observations.</p>
<p>The Minimum Covariance Determinant estimator is a robust,
high-breakdown point (i.e. it can be used to estimate the covariance
matrix of highly contaminated datasets, up to
<span class="math notranslate nohighlight">\(\frac{n_\text{samples}-n_\text{features}-1}{2}\)</span> outliers)
estimator of covariance. The idea is to find
<span class="math notranslate nohighlight">\(\frac{n_\text{samples}+n_\text{features}+1}{2}\)</span>
observations whose empirical covariance has the smallest determinant,
yielding a “pure” subset of observations from which to compute
standards estimates of location and covariance.</p>
<p>The Minimum Covariance Determinant estimator (MCD) has been introduced
by P.J.Rousseuw in [1].</p>
<p>This example illustrates how the Mahalanobis distances are affected by
outlying data: observations drawn from a contaminating distribution
are not distinguishable from the observations coming from the real,
Gaussian distribution that one may want to work with. Using MCD-based
Mahalanobis distances, the two populations become
distinguishable. Associated applications are outliers detection,
observations ranking, clustering, …
For visualization purpose, the cubic root of the Mahalanobis distances
are represented in the boxplot, as Wilson and Hilferty suggest [2]</p>
<dl class="simple">
<dt>[1] P. J. Rousseeuw. Least median of squares regression. J. Am</dt><dd><p>Stat Ass, 79:871, 1984.</p>
</dd>
<dt>[2] Wilson, E. B., &amp; Hilferty, M. M. (1931). The distribution of chi-square.</dt><dd><p>Proceedings of the National Academy of Sciences of the United States
of America, 17, 684-688.</p>
</dd>
</dl>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>

<span class="kn">from</span> <span class="nn">sklearn.covariance</span> <span class="kn">import</span> <span class="n">EmpiricalCovariance</span><span class="p">,</span> <span class="n">MinCovDet</span>

<span class="n">n_samples</span> <span class="o">=</span> <span class="mi">125</span>
<span class="n">n_outliers</span> <span class="o">=</span> <span class="mi">25</span>
<span class="n">n_features</span> <span class="o">=</span> <span class="mi">2</span>

<span class="c1"># generate data</span>
<span class="n">gen_cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span>
<span class="n">gen_cov</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mf">2.</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_features</span><span class="p">),</span> <span class="n">gen_cov</span><span class="p">)</span>
<span class="c1"># add some outliers</span>
<span class="n">outliers_cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n_features</span><span class="p">)</span>
<span class="n">outliers_cov</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_features</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n_features</span><span class="p">)]</span> <span class="o">=</span> <span class="mf">7.</span>
<span class="n">X</span><span class="p">[</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="n">n_outliers</span><span class="p">,</span> <span class="n">n_features</span><span class="p">),</span> <span class="n">outliers_cov</span><span class="p">)</span>

<span class="c1"># fit a Minimum Covariance Determinant (MCD) robust estimator to data</span>
<span class="n">robust_cov</span> <span class="o">=</span> <span class="n">MinCovDet</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

<span class="c1"># compare estimators learnt from the full data set with true parameters</span>
<span class="n">emp_cov</span> <span class="o">=</span> <span class="n">EmpiricalCovariance</span><span class="p">()</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># Display results</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">hspace</span><span class="o">=-.</span><span class="mi">1</span><span class="p">,</span> <span class="n">wspace</span><span class="o">=.</span><span class="mi">4</span><span class="p">,</span> <span class="n">top</span><span class="o">=.</span><span class="mi">95</span><span class="p">,</span> <span class="n">bottom</span><span class="o">=.</span><span class="mi">05</span><span class="p">)</span>

<span class="c1"># Show data set</span>
<span class="n">subfig1</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">inlier_plot</span> <span class="o">=</span> <span class="n">subfig1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span>
                              <span class="n">color</span><span class="o">=</span><span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;inliers&#39;</span><span class="p">)</span>
<span class="n">outlier_plot</span> <span class="o">=</span> <span class="n">subfig1</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">][</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:],</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">][</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:],</span>
                               <span class="n">color</span><span class="o">=</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">&#39;outliers&#39;</span><span class="p">)</span>
<span class="n">subfig1</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="n">subfig1</span><span class="o">.</span><span class="n">get_xlim</span><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="mf">11.</span><span class="p">)</span>
<span class="n">subfig1</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;Mahalanobis distances of a contaminated data set:&quot;</span><span class="p">)</span>

<span class="c1"># Show contours of the distance functions</span>
<span class="n">xx</span><span class="p">,</span> <span class="n">yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">()[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">100</span><span class="p">),</span>
                     <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">()[</span><span class="mi">0</span><span class="p">],</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">()[</span><span class="mi">1</span><span class="p">],</span> <span class="mi">100</span><span class="p">))</span>
<span class="n">zz</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">xx</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">yy</span><span class="o">.</span><span class="n">ravel</span><span class="p">()]</span>

<span class="n">mahal_emp_cov</span> <span class="o">=</span> <span class="n">emp_cov</span><span class="o">.</span><span class="n">mahalanobis</span><span class="p">(</span><span class="n">zz</span><span class="p">)</span>
<span class="n">mahal_emp_cov</span> <span class="o">=</span> <span class="n">mahal_emp_cov</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xx</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">emp_cov_contour</span> <span class="o">=</span> <span class="n">subfig1</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mahal_emp_cov</span><span class="p">),</span>
                                  <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">PuBu_r</span><span class="p">,</span>
                                  <span class="n">linestyles</span><span class="o">=</span><span class="s1">&#39;dashed&#39;</span><span class="p">)</span>

<span class="n">mahal_robust_cov</span> <span class="o">=</span> <span class="n">robust_cov</span><span class="o">.</span><span class="n">mahalanobis</span><span class="p">(</span><span class="n">zz</span><span class="p">)</span>
<span class="n">mahal_robust_cov</span> <span class="o">=</span> <span class="n">mahal_robust_cov</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">xx</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">robust_contour</span> <span class="o">=</span> <span class="n">subfig1</span><span class="o">.</span><span class="n">contour</span><span class="p">(</span><span class="n">xx</span><span class="p">,</span> <span class="n">yy</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">mahal_robust_cov</span><span class="p">),</span>
                                 <span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">YlOrBr_r</span><span class="p">,</span> <span class="n">linestyles</span><span class="o">=</span><span class="s1">&#39;dotted&#39;</span><span class="p">)</span>

<span class="n">subfig1</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="n">emp_cov_contour</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">robust_contour</span><span class="o">.</span><span class="n">collections</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span>
                <span class="n">inlier_plot</span><span class="p">,</span> <span class="n">outlier_plot</span><span class="p">],</span>
               <span class="p">[</span><span class="s1">&#39;MLE dist&#39;</span><span class="p">,</span> <span class="s1">&#39;robust dist&#39;</span><span class="p">,</span> <span class="s1">&#39;inliers&#39;</span><span class="p">,</span> <span class="s1">&#39;outliers&#39;</span><span class="p">],</span>
               <span class="n">loc</span><span class="o">=</span><span class="s2">&quot;upper right&quot;</span><span class="p">,</span> <span class="n">borderaxespad</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>

<span class="c1"># Plot the scores for each point</span>
<span class="n">emp_mahal</span> <span class="o">=</span> <span class="n">emp_cov</span><span class="o">.</span><span class="n">mahalanobis</span><span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span> <span class="o">**</span> <span class="p">(</span><span class="mf">0.33</span><span class="p">)</span>
<span class="n">subfig2</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">subfig2</span><span class="o">.</span><span class="n">boxplot</span><span class="p">([</span><span class="n">emp_mahal</span><span class="p">[:</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">],</span> <span class="n">emp_mahal</span><span class="p">[</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:]],</span> <span class="n">widths</span><span class="o">=.</span><span class="mi">25</span><span class="p">)</span>
<span class="n">subfig2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">n_samples</span> <span class="o">-</span> <span class="n">n_outliers</span><span class="p">,</span> <span class="mf">1.26</span><span class="p">),</span>
             <span class="n">emp_mahal</span><span class="p">[:</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">],</span> <span class="s1">&#39;+k&#39;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">subfig2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">n_outliers</span><span class="p">,</span> <span class="mf">2.26</span><span class="p">),</span>
             <span class="n">emp_mahal</span><span class="p">[</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:],</span> <span class="s1">&#39;+k&#39;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">subfig2</span><span class="o">.</span><span class="n">axes</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">((</span><span class="s1">&#39;inliers&#39;</span><span class="p">,</span> <span class="s1">&#39;outliers&#39;</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">subfig2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$\sqrt[3]{\rm{(Mahal. dist.)}}$&quot;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
<span class="n">subfig2</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;1. from non-robust estimates</span><span class="se">\n</span><span class="s2">(Maximum Likelihood)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>

<span class="n">robust_mahal</span> <span class="o">=</span> <span class="n">robust_cov</span><span class="o">.</span><span class="n">mahalanobis</span><span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="n">robust_cov</span><span class="o">.</span><span class="n">location_</span><span class="p">)</span> <span class="o">**</span> <span class="p">(</span><span class="mf">0.33</span><span class="p">)</span>
<span class="n">subfig3</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">subfig3</span><span class="o">.</span><span class="n">boxplot</span><span class="p">([</span><span class="n">robust_mahal</span><span class="p">[:</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">],</span> <span class="n">robust_mahal</span><span class="p">[</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:]],</span>
                <span class="n">widths</span><span class="o">=.</span><span class="mi">25</span><span class="p">)</span>
<span class="n">subfig3</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">n_samples</span> <span class="o">-</span> <span class="n">n_outliers</span><span class="p">,</span> <span class="mf">1.26</span><span class="p">),</span>
             <span class="n">robust_mahal</span><span class="p">[:</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">],</span> <span class="s1">&#39;+k&#39;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">subfig3</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">full</span><span class="p">(</span><span class="n">n_outliers</span><span class="p">,</span> <span class="mf">2.26</span><span class="p">),</span>
             <span class="n">robust_mahal</span><span class="p">[</span><span class="o">-</span><span class="n">n_outliers</span><span class="p">:],</span> <span class="s1">&#39;+k&#39;</span><span class="p">,</span> <span class="n">markeredgewidth</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">subfig3</span><span class="o">.</span><span class="n">axes</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">((</span><span class="s1">&#39;inliers&#39;</span><span class="p">,</span> <span class="s1">&#39;outliers&#39;</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
<span class="n">subfig3</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="sa">r</span><span class="s2">&quot;$\sqrt[3]{\rm{(Mahal. dist.)}}$&quot;</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="mi">16</span><span class="p">)</span>
<span class="n">subfig3</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="s2">&quot;2. from robust estimates</span><span class="se">\n</span><span class="s2">(Minimum Covariance Determinant)&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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